This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem fabex

Description: Existence of a set of functions. (Contributed by NM, 3-Dec-2007)

Ref Expression
Hypotheses fabex.1 ⊢ 𝐴 ∈ V
fabex.2 ⊢ 𝐵 ∈ V
fabex.3 ⊢ 𝐹 = { 𝑥 ∣ ( 𝑥 : 𝐴 ⟶ 𝐵 ∧ 𝜑 ) }
Assertion fabex 𝐹 ∈ V

Proof

Step Hyp Ref Expression
1 fabex.1 ⊢ 𝐴 ∈ V
2 fabex.2 ⊢ 𝐵 ∈ V
3 fabex.3 ⊢ 𝐹 = { 𝑥 ∣ ( 𝑥 : 𝐴 ⟶ 𝐵 ∧ 𝜑 ) }
4 3 fabexg ⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → 𝐹 ∈ V )
5 1 2 4 mp2an ⊢ 𝐹 ∈ V